Hierarchical testing using the power-divergence family of statistics

Methodology for discrete multivariate data based on the loglikelihood ratio statistic G[superscript]2, and Pearson\u27s statistic X[superscript]2 is extended to the power-divergence family of goodness-of-fit statistics (Cressie and Read, 1984), which is indexed by the parameter [lambda] (-[infinity] \u3c [lambda] \u3c [infinity]). This family includes G[superscript]2, X[superscript]2, the Freeman-Tukey statistic, the modified loglikelihood ratio statistic, and the Neyman-modified chi-squared statistic;Ideas employed by Watson and Nguyen (1985) and Watson (1987) to plot confidence regions in a ternary diagram, based on Pearson\u27s X[superscript]2, are extended to the power-divergence family. This results in confidence regions of diverse shapes and sizes. Also, a comparison based on the accuracy of confidence level and the area of confidence region finds the family members [lambda] = 2/3 and [lambda] = 1/2 to be the best performers;Maximum likelihood methods (e.g., Bishop, Fienberg, and Holland, 1975, Chapters 4 and 14) for testing hierarchical parametric models are extended to the power-divergence family. It is shown that, under Birch\u27s conditions (Birch, 1964), an analysis of divergence is possible with the power-divergence family, analogous to the usual partitioning of G[superscript]2 given, e.g., in Fienberg (1980, pp. 58-59). Further, an algorithm similar to iterative proportional fitting, for finding cell probability estimates, is given. To illustrate these ideas loglinear models are fit to several data sets and analyses of divergence are carried out;Methodology for hierarchically assessing homogeneity in product-multinomial distributions, based on the power-divergence statistics, is developed. It is shown that, under mild assumptions, an analysis of divergence for the power-divergence statistics is possible. To demonstrate this methodology, a data set is considered and an analysis of divergence is performed